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Properties

Label 4-1680e2-1.1-c1e2-0-1
Degree 44
Conductor 28224002822400
Sign 11
Analytic cond. 179.958179.958
Root an. cond. 3.662633.66263
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 4·7-s − 11-s + 2·13-s − 15-s − 3·19-s + 4·21-s + 7·23-s + 27-s − 16·29-s − 2·31-s + 33-s − 4·35-s − 11·37-s − 2·39-s − 22·41-s − 16·43-s − 5·47-s + 9·49-s + 11·53-s − 55-s + 3·57-s + 4·59-s + 2·65-s − 7·69-s + 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.688·19-s + 0.872·21-s + 1.45·23-s + 0.192·27-s − 2.97·29-s − 0.359·31-s + 0.174·33-s − 0.676·35-s − 1.80·37-s − 0.320·39-s − 3.43·41-s − 2.43·43-s − 0.729·47-s + 9/7·49-s + 1.51·53-s − 0.134·55-s + 0.397·57-s + 0.520·59-s + 0.248·65-s − 0.842·69-s + 1.42·71-s + ⋯

Functional equation

Λ(s)=(2822400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2822400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 28224002822400    =    28⋅32⋅52⋅722^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 179.958179.958
Root analytic conductor: 3.662633.66263
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2822400, ( :1/2,1/2), 1)(4,\ 2822400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.048414158950.04841415895
L(12)L(\frac12) ≈\approx 0.048414158950.04841415895
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C2C_2 1+T+T2 1 + T + T^{2}
5C2C_2 1−T+T2 1 - T + T^{2}
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good11C22C_2^2 1+T−10T2+pT3+p2T4 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4}
13C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C22C_2^2 1+3T−10T2+3pT3+p2T4 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−7T+26T2−7pT3+p2T4 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4}
29C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
31C22C_2^2 1+2T−27T2+2pT3+p2T4 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4}
37C2C_2 (1+T+pT2)(1+10T+pT2) ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C22C_2^2 1+5T−22T2+5pT3+p2T4 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−11T+68T2−11pT3+p2T4 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−4T−43T2−4pT3+p2T4 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
67C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C22C_2^2 1−6T−37T2−6pT3+p2T4 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+8T−15T2+8pT3+p2T4 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
89C22C_2^2 1−10T+11T2−10pT3+p2T4 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4}
97C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.645529194206775674237179521887, −9.309689410272044474232605311289, −8.666328106996505670956017161449, −8.363259098663911736200420303808, −8.312083123535375178383890198474, −7.08918743127239478044254391619, −7.07423613665469414586408719283, −6.89001979491523585099234649437, −6.41807598244263102116197279392, −5.70158173067212268174121192934, −5.67792637778453247867409500125, −5.04168140523126230106896089692, −4.92545889399787184421972231256, −3.78222559471431756922802333872, −3.64969016491551752799892052416, −3.30921537606468101755556365988, −2.63533721770063924908287739903, −1.84697273236623692140284270232, −1.49137808468798994103135524713, −0.084000489316477839434140689195, 0.084000489316477839434140689195, 1.49137808468798994103135524713, 1.84697273236623692140284270232, 2.63533721770063924908287739903, 3.30921537606468101755556365988, 3.64969016491551752799892052416, 3.78222559471431756922802333872, 4.92545889399787184421972231256, 5.04168140523126230106896089692, 5.67792637778453247867409500125, 5.70158173067212268174121192934, 6.41807598244263102116197279392, 6.89001979491523585099234649437, 7.07423613665469414586408719283, 7.08918743127239478044254391619, 8.312083123535375178383890198474, 8.363259098663911736200420303808, 8.666328106996505670956017161449, 9.309689410272044474232605311289, 9.645529194206775674237179521887

Graph of the ZZ-function along the critical line